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Chapter 11 Class 12 Three Dimensional Geometry (Term 2)

Serial order wise

Last updated at Feb. 4, 2020 by Teachoo

Example 30 (Method 1) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points with position vectors π β & π β is π β = π β + π (π β β π β) Given the line passes through the points (π β β π β) = (5π Μ + 1π Μ + 6π Μ) β (3π Μ + 4π Μ + 1π Μ) = 2π Μ β 3π Μ + 5π Μ A (3, 4, 1) π β = 3π Μ + 4π Μ + π Μ B (5, 1, 6) π β = 5π Μ + 1π Μ + 6π Μ β΄ π β = (3π Μ + 4π Μ + 1π Μ) + π (2π Μ β 3π Μ + 5π Μ) Let the coordinates of the point where the line crosses the XY plane be (x, y, 0). So, π β = xπ Μ + yπ Μ + 0π Μ Since point crosses the plane, it will satisfy its equation Putting (2) in (1) xπ Μ + yπ Μ + 0π Μ = 3π Μ + 4π Μ + 1π Μ + 2ππ Μ β 3ππ Μ + 5ππ Μ xπ Μ + yπ Μ + 0π Μ = (3 + 2π)π Μ + (4 β 3π)π Μ + (1 + 5π)π Μ Two vectors are equal if their corresponding components are equal So, Solving 0 = 1 + 5π β΄ π = (βπ)/π So, x = 3 + π = 3 + 2 Γ (β1)/5 = 3 β 2/5 = 13/5 & y = 4 β 3π = 4 β 3 Γ (β1)/5 = 4 + 3/5 = 23/5 Therefore, the required coordinates are (ππ/π,ππ/π,π) Example 30 (Method 2) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points A(π₯_1, π¦_1, π§_1) and B(π₯_2, π¦_2, π§_2) is (π β π_π)/(π_π β π_π ) = (π β π_π)/(π_π β π_π ) = (π β π_π)/(π_π β π_π ) Given the line passes through the points So, the equation of line is (π₯ β 3)/(5 β 3) = (π¦ β 4)/(1 β 4) = (π§ β 1)/(6 β 1) A (3, 4, 1) β΄ π₯_1= 3, π¦_1= 4, π§_1= 1 B(5, 1, 6) β΄ π₯_2 = 5, π¦_2= 1, π§_2= 6 (π₯ β 3)/2 = (π¦ β 4)/(β3) = (π§ β 1)/5 = k So, Since, the line crosses the XY plane at (x, y, 0) z = 0 5k + 1 = 0 5k = β1 β΄ k = (βπ)/π So, x = 2k + 3 = 2 Γ (β1)/5 + 3 = 3 β 2/5 = 13/5 y = β3 Γ (β1)/5 + 4 = 4 + 3/5 = 23/5 Therefore, the coordinates of the required point are (ππ/π, ππ/π, π).